:: Miscellaneous Facts about Open Functions and Continuous Functions
:: by Artur Korni{\l}owicz
::
:: Received February 9, 2010
:: Copyright (c) 2010 Association of Mizar Users
environ
vocabularies RELAT_1, FUNCT_1, TOPS_1, ARYTM_1, CONNSP_2, EUCLID, PRE_TOPC,
ORDINAL2, METRIC_1, TOPMETR, RCOMP_1, PCOMPS_1, TARSKI, XBOOLE_0,
NUMBERS, NAT_1, SUBSET_1, XXREAL_0, ARYTM_3, ORDINAL1, XREAL_0, CARD_1,
XXREAL_1;
notations TARSKI, SUBSET_1, FUNCT_1, RELSET_1, FUNCT_2, ORDINAL1, NUMBERS,
XCMPLX_0, XREAL_0, XXREAL_0, MEMBERED, NAT_1, RCOMP_1, STRUCT_0,
PRE_TOPC, TOPS_1, CONNSP_2, METRIC_1, PCOMPS_1, TOPMETR, T_0TOPSP,
EUCLID, TOPREAL9, TOPREALB;
constructors TOPREAL9, TOPREALB, TOPS_1, BORSUK_4, T_0TOPSP, FUNCSDOM,
PCOMPS_1;
registrations XBOOLE_0, RELAT_1, MEMBERED, XREAL_0, PRE_TOPC, STRUCT_0,
EUCLID, TOPREALB, XXREAL_0, METRIC_1, BORSUK_6, TOPMETR, TOPREAL9,
PCOMPS_1, RCOMP_1, TOPGRP_1, EUCLID_9;
requirements SUBSET, REAL;
definitions TARSKI, T_0TOPSP, METRIC_1, PCOMPS_1, TOPMETR, BORSUK_4, TOPREALB;
theorems FUNCT_2, TOPMETR, TOPREAL9, EUCLID, JGRAPH_2, GOBOARD6, TOPS_1,
METRIC_1, XBOOLE_1, T_0TOPSP, XREAL_0, FRECHET, TOPREAL6, RELAT_1,
UNIFORM1, CONNSP_2, TOPGRP_1, ORDINAL1, TOPS_3, YELLOW12, TOPGEN_5;
begin :: Open functions
reserve
n, m for Nat,
T for non empty TopSpace,
M, M1, M2 for non empty MetrSpace;
theorem Th1:
for A, B, S, T being TopSpace,
f being Function of A,S, g being Function of B,T st
the TopStruct of A = the TopStruct of B &
the TopStruct of S = the TopStruct of T &
f = g & f is open holds g is open
proof
let A, B, S, T be TopSpace;
let f be Function of A,S;
let g be Function of B,T;
assume that
A1: the TopStruct of A = the TopStruct of B and
A2: the TopStruct of S = the TopStruct of T and
A3: f = g and
A4: f is open;
let b be Subset of B;
assume
A5: b is open;
reconsider a = b as Subset of A by A1;
a is open by A1,A5,TOPS_3:76;
then f.:a is open by A4,T_0TOPSP:def 2;
hence thesis by A2,A3,TOPS_3:76;
end;
theorem
for P being Subset of TOP-REAL m holds P is open iff
for p being Point of TOP-REAL m st p in P
ex r being positive (real number) st Ball(p,r) c= P
proof
let P be Subset of TOP-REAL m;
A1: the TopStruct of TOP-REAL m = TopSpaceMetr Euclid m by EUCLID:def 8;
then
reconsider P1 = P as Subset of TopSpaceMetr(Euclid m);
A2: m in NAT by ORDINAL1:def 13;
hereby
assume
A3: P is open;
let p be Point of TOP-REAL m;
assume
A4: p in P;
reconsider e = p as Point of Euclid m by EUCLID:71;
P1 is open by A3,A1,TOPS_3:76;
then consider r being real number such that
A5: r > 0 and
A6: Ball(e,r) c= P1 by A4,TOPMETR:22;
reconsider r as positive (real number) by A5;
take r;
thus Ball(p,r) c= P by A2,A6,TOPREAL9:13;
end;
assume
A7: for p being Point of TOP-REAL m st p in P
ex r being positive (real number) st Ball(p,r) c= P;
for p being Point of Euclid m st p in P1
ex r being real number st r > 0 & Ball(p,r) c= P1
proof
let p be Point of Euclid m;
assume
A8: p in P1;
reconsider e = p as Point of TOP-REAL m by EUCLID:71;
consider r being positive (real number) such that
A9: Ball(e,r) c= P1 by A7,A8;
take r;
thus thesis by A2,A9,TOPREAL9:13;
end;
then P1 is open by TOPMETR:22;
hence thesis by A1,TOPS_3:76;
end;
theorem Th3:
for X, Y being non empty TopSpace, f being Function of X,Y
holds f is open iff for p being Point of X, V being open Subset of X
st p in V
ex W being open Subset of Y st f.p in W & W c= f.:V
proof
let X, Y be non empty TopSpace, f be Function of X,Y;
thus f is open implies for p being Point of X, V being open Subset of X
st p in V
ex W being open Subset of Y st f.p in W & W c= f.:V
proof
assume
A1: f is open;
let p be Point of X, V be open Subset of X such that
A2: p in V;
V is a_neighborhood of p by A2,CONNSP_2:5;
then consider W being open a_neighborhood of f.p such that
A3: W c= f.:V by A1,TOPGRP_1:22;
take W;
thus f.p in W by CONNSP_2:6;
thus thesis by A3;
end;
assume
A4: for p being Point of X, V being open Subset of X st p in V
ex W being open Subset of Y st f.p in W & W c= f.:V;
for p being Point of X, P being open a_neighborhood of p
ex R being a_neighborhood of f.p st R c= f.:P
proof
let p be Point of X;
let P be open a_neighborhood of p;
consider W being open Subset of Y such that
A5: f.p in W and
A6: W c= f.:P by A4,CONNSP_2:6;
W is a_neighborhood of f.p by A5,CONNSP_2:5;
hence thesis by A6;
end;
hence thesis by TOPGRP_1:23;
end;
theorem Th4:
for f being Function of T,TopSpaceMetr(M) holds f is open iff
for p being Point of T, V being open Subset of T,
q being Point of M st q = f.p & p in V
ex r being positive (real number) st Ball(q,r) c= f.:V
proof
let f be Function of T,TopSpaceMetr(M);
thus f is open implies for p being Point of T, V being open Subset of T,
q being Point of M st q = f.p & p in V
ex r being positive (real number) st Ball(q,r) c= f.:V
proof
assume
A1: f is open;
let p be Point of T, V be open Subset of T,
q be Point of M such that
A2: q = f.p;
assume p in V;
then consider W being open Subset of TopSpaceMetr(M) such that
A3: f.p in W and
A4: W c= f.:V by A1,Th3;
ex r being real number st r > 0 & Ball(q,r) c= W by A2,A3,TOPMETR:22;
hence thesis by A4,XBOOLE_1:1;
end;
assume
A5: for p being Point of T, V being open Subset of T,
q being Point of M st q = f.p & p in V
ex r being positive (real number) st Ball(q,r) c= f.:V;
for p being Point of T, V being open Subset of T st p in V
ex W being open Subset of TopSpaceMetr(M) st f.p in W & W c= f.:V
proof
let p be Point of T;
let V be open Subset of T;
reconsider q = f.p as Point of M;
assume p in V;
then consider r being positive (real number) such that
A6: Ball(q,r) c= f.:V by A5;
reconsider W = Ball(q,r) as open Subset of TopSpaceMetr(M) by TOPMETR:21;
take W;
thus f.p in W by GOBOARD6:4;
thus W c= f.:V by A6;
end;
hence thesis by Th3;
end;
theorem Th5:
for f being Function of TopSpaceMetr(M),T holds f is open iff
for p being Point of M, r being positive (real number)
ex W being open Subset of T st f.p in W & W c= f.:Ball(p,r)
proof
let f be Function of TopSpaceMetr(M),T;
hereby
assume
A1: f is open;
let p be Point of M, r be positive (real number);
A2: p in Ball(p,r) by GOBOARD6:4;
reconsider V = Ball(p,r) as Subset of TopSpaceMetr(M);
V is open by TOPMETR:21;
then consider W being open Subset of T such that
A3: f.p in W & W c= f.:V by A1,A2,Th3;
take W;
thus f.p in W & W c= f.:Ball(p,r) by A3;
end;
assume
A4: for p being Point of M, r being positive (real number)
ex W being open Subset of T st f.p in W & W c= f.:Ball(p,r);
for p being Point of TopSpaceMetr(M),
V being open Subset of TopSpaceMetr(M) st p in V
ex W being open Subset of T st f.p in W & W c= f.:V
proof
let p be Point of TopSpaceMetr(M);
let V be open Subset of TopSpaceMetr(M);
reconsider q = p as Point of M;
assume p in V;
then consider r being real number such that
A5: r > 0 and
A6: Ball(q,r) c= V by TOPMETR:22;
consider W being open Subset of T such that
A7: f.p in W and
A8: W c= f.:Ball(q,r) by A4,A5;
take W;
thus f.p in W by A7;
f.:Ball(q,r) c= f.:V by A6,RELAT_1:156;
hence thesis by A8,XBOOLE_1:1;
end;
hence thesis by Th3;
end;
theorem Th6:
for f being Function of TopSpaceMetr(M1),TopSpaceMetr(M2) holds f is open iff
for p being Point of M1, q being Point of M2, r being positive (real number)
st q = f.p
ex s being positive (real number) st Ball(q,s) c= f.:Ball(p,r)
proof
let f be Function of TopSpaceMetr(M1),TopSpaceMetr(M2);
thus f is open implies for p being Point of M1, q being Point of M2,
r being positive (real number) st q = f.p
ex s being positive (real number) st Ball(q,s) c= f.:Ball(p,r)
proof
assume
A1: f is open;
let p be Point of M1, q be Point of M2, r be positive (real number)
such that
A2: q = f.p;
reconsider V = Ball(p,r) as Subset of TopSpaceMetr(M1);
A3: p in V by GOBOARD6:4;
V is open by TOPMETR:21;
hence thesis by A1,A2,A3,Th4;
end;
assume
A4: for p being Point of M1, q being Point of M2,
r being positive (real number) st q = f.p
ex s being positive (real number) st Ball(q,s) c= f.:Ball(p,r);
for p being Point of TopSpaceMetr(M1),
V being open Subset of TopSpaceMetr(M1),
q being Point of M2 st q = f.p & p in V
ex r being positive (real number) st Ball(q,r) c= f.:V
proof
let p be Point of TopSpaceMetr(M1), V be open Subset of TopSpaceMetr(M1),
q be Point of M2 such that
A5: q = f.p;
reconsider p1 = p as Point of M1;
assume p in V;
then consider r being real number such that
A6: r > 0 and
A7: Ball(p1,r) c= V by TOPMETR:22;
A8: f.:Ball(p1,r) c= f.:V by A7,RELAT_1:156;
ex s being positive (real number) st Ball(q,s) c= f.:Ball(p1,r)
by A4,A5,A6;
hence thesis by A8,XBOOLE_1:1;
end;
hence thesis by Th4;
end;
theorem
for f being Function of T,TOP-REAL m holds f is open iff
for p being Point of T, V being open Subset of T st p in V
ex r being positive (real number) st Ball(f.p,r) c= f.:V
proof
let f be Function of T,TOP-REAL m;
A1: the TopStruct of TOP-REAL m = TopSpaceMetr Euclid m by EUCLID:def 8;
then reconsider f1 = f as Function of T,TopSpaceMetr Euclid m;
A2: the TopStruct of T = the TopStruct of T;
A3: m in NAT by ORDINAL1:def 13;
thus f is open implies
for p being Point of T, V being open Subset of T st p in V
ex r being positive (real number) st Ball(f.p,r) c= f.:V
proof
assume
A4: f is open;
let p be Point of T, V be open Subset of T;
assume
A5: p in V;
reconsider fp = f.p as Point of Euclid m by EUCLID:71;
f1 is open by A4,A1,A2,Th1;
then consider r being positive (real number) such that
A6: Ball(fp,r) c= f1.:V by A5,Th4;
Ball(f.p,r) = Ball(fp,r) by A3,TOPREAL9:13;
hence thesis by A6;
end;
assume
A7: for p being Point of T, V being open Subset of T st p in V
ex r being positive (real number) st Ball(f.p,r) c= f.:V;
for p being Point of T, V being open Subset of T,
q being Point of Euclid m st q = f1.p & p in V
ex r being positive (real number) st Ball(q,r) c= f1.:V
proof
let p be Point of T, V be open Subset of T,
q be Point of Euclid m such that
A8: q = f1.p;
assume p in V;
then consider r being positive (real number) such that
A9: Ball(f.p,r) c= f.:V by A7;
Ball(f.p,r) = Ball(q,r) by A3,A8,TOPREAL9:13;
hence thesis by A9;
end;
then f1 is open by Th4;
hence thesis by A1,A2,Th1;
end;
theorem
for f being Function of TOP-REAL m,T holds f is open iff
for p being Point of TOP-REAL m, r being positive (real number)
ex W being open Subset of T st f.p in W & W c= f.:Ball(p,r)
proof
let f be Function of TOP-REAL m,T;
A1: the TopStruct of TOP-REAL m = TopSpaceMetr Euclid m by EUCLID:def 8;
then reconsider f1 = f as Function of TopSpaceMetr Euclid m,T;
A2: the TopStruct of T = the TopStruct of T;
A3: m in NAT by ORDINAL1:def 13;
thus f is open implies
for p being Point of TOP-REAL m, r being positive (real number)
ex W being open Subset of T st f.p in W & W c= f.:Ball(p,r)
proof
assume
A4: f is open;
let p be Point of TOP-REAL m, r be positive (real number);
reconsider q = p as Point of Euclid m by EUCLID:71;
f1 is open by A4,A1,A2,Th1;
then consider W be open Subset of T such that
A5: f1.p in W & W c= f1.:Ball(q,r) by Th5;
Ball(p,r) = Ball(q,r) by A3,TOPREAL9:13;
hence thesis by A5;
end;
assume
A6: for p being Point of TOP-REAL m, r being positive (real number)
ex W being open Subset of T st f.p in W & W c= f.:Ball(p,r);
for p being Point of Euclid m, r being positive (real number)
ex W being open Subset of T st f1.p in W & W c= f1.:Ball(p,r)
proof
let p be Point of Euclid m, r be positive (real number);
reconsider q = p as Point of TOP-REAL m by EUCLID:71;
consider W being open Subset of T such that
A7: f.q in W & W c= f.:Ball(q,r) by A6;
Ball(p,r) = Ball(q,r) by A3,TOPREAL9:13;
hence thesis by A7;
end;
then f1 is open by Th5;
hence thesis by A1,A2,Th1;
end;
theorem
for f being Function of TOP-REAL m,TOP-REAL n holds f is open iff
for p being Point of TOP-REAL m, r being positive (real number)
ex s being positive (real number) st Ball(f.p,s) c= f.:Ball(p,r)
proof
let f be Function of TOP-REAL m,TOP-REAL n;
A1: the TopStruct of TOP-REAL m = TopSpaceMetr Euclid m &
the TopStruct of TOP-REAL n = TopSpaceMetr Euclid n by EUCLID:def 8;
then reconsider f1 = f as
Function of TopSpaceMetr Euclid m,TopSpaceMetr Euclid n;
A2: m in NAT & n in NAT by ORDINAL1:def 13;
thus f is open implies
for p being Point of TOP-REAL m, r being positive (real number)
ex s being positive (real number) st Ball(f.p,s) c= f.:Ball(p,r)
proof
assume
A3: f is open;
let p be Point of TOP-REAL m, r be positive (real number);
reconsider p1 = p as Point of Euclid m by EUCLID:71;
reconsider q1 = f.p as Point of Euclid n by EUCLID:71;
f1 is open by A1,A3,Th1;
then consider s being positive (real number) such that
A4: Ball(q1,s) c= f1.:Ball(p1,r) by Th6;
Ball(p1,r) = Ball(p,r) & Ball(q1,s) = Ball(f.p,s) by A2,TOPREAL9:13;
hence thesis by A4;
end;
assume
A5: for p being Point of TOP-REAL m, r being positive (real number)
ex s being positive (real number) st Ball(f.p,s) c= f.:Ball(p,r);
for p being Point of Euclid m, q being Point of Euclid n,
r being positive (real number) st q = f1.p
ex s being positive (real number) st Ball(q,s) c= f1.:Ball(p,r)
proof
let p be Point of Euclid m, q be Point of Euclid n,
r be positive (real number) such that
A6: q = f1.p;
reconsider p1 = p as Point of TOP-REAL m by EUCLID:71;
reconsider q1 = q as Point of TOP-REAL n by EUCLID:71;
consider s being positive (real number) such that
A7: Ball(q1,s) c= f.:Ball(p1,r) by A5,A6;
Ball(p1,r) = Ball(p,r) & Ball(q1,s) = Ball(q,s) by A2,TOPREAL9:13;
hence thesis by A7;
end;
then f1 is open by Th6;
hence thesis by A1,Th1;
end;
theorem
for f being Function of T,R^1 holds f is open iff
for p being Point of T, V being open Subset of T st p in V
ex r being positive (real number) st ].f.p-r,f.p+r.[ c= f.:V
proof
let f be Function of T,R^1;
thus f is open implies
for p being Point of T, V being open Subset of T st p in V
ex r being positive (real number) st ].f.p-r,f.p+r.[ c= f.:V
proof
assume
A1: f is open;
let p be Point of T, V be open Subset of T;
assume
A2: p in V;
reconsider fp = f.p as Point of RealSpace;
consider r being positive (real number) such that
A3: Ball(fp,r) c= f.:V by A1,A2,Th4;
r in REAL by XREAL_0:def 1;
then ].fp-r,fp+r.[ = Ball(fp,r) by FRECHET:7;
hence thesis by A3;
end;
assume
A4: for p being Point of T, V being open Subset of T st p in V
ex r being positive (real number) st ].f.p-r,f.p+r.[ c= f.:V;
for p being Point of T, V being open Subset of T,
q being Point of RealSpace st q = f.p & p in V
ex r being positive (real number) st Ball(q,r) c= f.:V
proof
let p be Point of T, V be open Subset of T,
q be Point of RealSpace such that
A5: q = f.p;
assume p in V;
then consider r being positive (real number) such that
A6: ].f.p-r,f.p+r.[ c= f.:V by A4;
r in REAL by XREAL_0:def 1;
then ].q-r,q+r.[ = Ball(q,r) by FRECHET:7;
hence thesis by A5,A6;
end;
hence thesis by Th4;
end;
theorem
for f being Function of R^1,T holds f is open iff
for p being Point of R^1, r being positive (real number)
ex V being open Subset of T st f.p in V & V c= f.:].p-r,p+r.[
proof
let f be Function of R^1,T;
thus f is open implies
for p being Point of R^1, r being positive (real number)
ex W being open Subset of T st f.p in W & W c= f.:].p-r,p+r.[
proof
assume
A1: f is open;
let p be Point of R^1, r be positive (real number);
reconsider q = p as Point of RealSpace;
consider W be open Subset of T such that
A2: f.p in W & W c= f.:Ball(q,r) by A1,Th5;
r in REAL by XREAL_0:def 1;
then ].q-r,q+r.[ = Ball(q,r) by FRECHET:7;
hence thesis by A2;
end;
assume
A3: for p being Point of R^1, r being positive (real number)
ex W being open Subset of T st f.p in W & W c= f.:].p-r,p+r.[;
for p being Point of RealSpace, r being positive (real number)
ex W being open Subset of T st f.p in W & W c= f.:Ball(p,r)
proof
let p be Point of RealSpace, r be positive (real number);
reconsider q = p as Point of R^1;
consider W being open Subset of T such that
A4: f.q in W & W c= f.:].p-r,p+r.[ by A3;
r in REAL by XREAL_0:def 1;
then ].p-r,p+r.[ = Ball(p,r) by FRECHET:7;
hence thesis by A4;
end;
hence thesis by Th5;
end;
theorem
for f being Function of R^1,R^1 holds f is open iff
for p being Point of R^1, r being positive (real number)
ex s being positive (real number) st ].f.p-s,f.p+s.[ c= f.:].p-r,p+r.[
proof
let f be Function of R^1,R^1;
thus f is open implies
for p being Point of R^1, r being positive (real number)
ex s being positive (real number) st ].f.p-s,f.p+s.[ c= f.:].p-r,p+r.[
proof
assume
A1: f is open;
let p be Point of R^1, r be positive (real number);
reconsider p1 = p, q1 = f.p as Point of RealSpace;
consider s being positive (real number) such that
A2: Ball(q1,s) c= f.:Ball(p1,r) by A1,Th6;
r in REAL & s in REAL by XREAL_0:def 1;
then ].p-r,p+r.[ = Ball(p1,r) & ].f.p-s,f.p+s.[ = Ball(q1,s)
by FRECHET:7;
hence thesis by A2;
end;
assume
A3: for p being Point of R^1, r being positive (real number)
ex s being positive (real number) st ].f.p-s,f.p+s.[ c= f.:].p-r,p+r.[;
for p, q being Point of RealSpace, r being positive (real number)
st q = f.p
ex s being positive (real number) st Ball(q,s) c= f.:Ball(p,r)
proof
let p, q be Point of RealSpace,
r be positive (real number) such that
A4: q = f.p;
consider s being positive (real number) such that
A5: ].f.p-s,f.p+s.[ c= f.:].p-r,p+r.[ by A3;
r in REAL & s in REAL by XREAL_0:def 1;
then ].p-r,p+r.[ = Ball(p,r) & ].f.p-s,f.p+s.[ = Ball(q,s)
by A4,FRECHET:7;
hence thesis by A5;
end;
hence thesis by Th6;
end;
theorem
for f being Function of TOP-REAL m,R^1 holds f is open iff
for p being Point of TOP-REAL m, r being positive (real number)
ex s being positive (real number) st ].f.p-s,f.p+s.[ c= f.:Ball(p,r)
proof
let f be Function of TOP-REAL m,R^1;
A1: m in NAT by ORDINAL1:def 13;
hereby
assume
A2: f is open;
let p be Point of TOP-REAL m, r be positive (real number);
p in Ball(p,r) by A1,TOPGEN_5:17;
then
A3: f.p in f.:Ball(p,r) by FUNCT_2:43;
f.:Ball(p,r) is open by A2,T_0TOPSP:def 2;
then consider s being Element of REAL such that
A4: s > 0 and
A5: ].f.p-s,f.p+s.[ c= f.:Ball(p,r) by A3,FRECHET:8;
reconsider s as positive (real number) by A4;
take s;
thus ].f.p-s,f.p+s.[ c= f.:Ball(p,r) by A5;
end;
assume
A6: for p being Point of TOP-REAL m, r being positive (real number)
ex s being positive (real number) st ].f.p-s,f.p+s.[ c= f.:Ball(p,r);
let A be Subset of TOP-REAL m such that
A7: A is open;
for x being set holds x in f.:A iff ex Q being Subset of R^1
st Q is open & Q c= f.:A & x in Q
proof
let x be set;
hereby
assume x in f.:A;
then consider p being Point of TOP-REAL m such that
A8: p in A and
A9: x = f.p by FUNCT_2:116;
reconsider u = p as Point of Euclid m by EUCLID:71;
A = Int A by A7,TOPS_1:55;
then consider e being real number such that
A10: e > 0 and
A11: Ball(u,e) c= A by A1,A8,GOBOARD6:8;
A12: Ball(u,e) = Ball(p,e) by A1,TOPREAL9:13;
consider s being positive (real number) such that
A13: ].f.p-s,f.p+s.[ c= f.:Ball(p,e) by A6,A10;
take Q = R^1(].f.p-s,f.p+s.[);
thus Q is open;
f.:Ball(p,e) c= f.:A by A11,A12,RELAT_1:156;
hence Q c= f.:A by A13,XBOOLE_1:1;
thus x in Q by A9,TOPREAL6:20;
end;
thus thesis;
end;
hence f.:A is open by TOPS_1:57;
end;
theorem
for f being Function of R^1,TOP-REAL m holds f is open iff
for p being Point of R^1, r being positive (real number)
ex s being positive (real number) st Ball(f.p,s) c= f.:].p-r,p+r.[
proof
let f be Function of R^1,TOP-REAL m;
A1: the TopStruct of TOP-REAL m = TopSpaceMetr Euclid m by EUCLID:def 8;
then reconsider f1 = f as Function of R^1,TopSpaceMetr Euclid m;
A2: m in NAT by ORDINAL1:def 13;
thus f is open implies
for p being Point of R^1, r being positive (real number)
ex s being positive (real number) st Ball(f.p,s) c= f.:].p-r,p+r.[
proof
assume
A3: f is open;
let p be Point of R^1, r be positive (real number);
reconsider p1 = p as Point of RealSpace;
reconsider q1 = f.p as Point of Euclid m by EUCLID:71;
f1 is open by A1,A3,Th1;
then consider s being positive (real number) such that
A4: Ball(q1,s) c= f1.:Ball(p1,r) by Th6;
r in REAL by XREAL_0:def 1;
then ].p-r,p+r.[ = Ball(p1,r) & Ball(f.p,s) = Ball(q1,s)
by A2,FRECHET:7,TOPREAL9:13;
hence thesis by A4;
end;
assume
A5: for p being Point of R^1, r being positive (real number)
ex s being positive (real number) st Ball(f.p,s) c= f.:].p-r,p+r.[;
for p being Point of RealSpace, q being Point of Euclid m,
r being positive (real number) st q = f1.p
ex s being positive (real number) st Ball(q,s) c= f1.:Ball(p,r)
proof
let p be Point of RealSpace, q be Point of Euclid m,
r be positive (real number) such that
A6: q = f1.p;
reconsider p1 = p as Point of R^1;
consider s being positive (real number) such that
A7: Ball(f.p1,s) c= f.:].p1-r,p1+r.[ by A5;
r in REAL & s in REAL by XREAL_0:def 1;
then ].p1-r,p1+r.[ = Ball(p,r) & Ball(f.p1,s) = Ball(q,s)
by A2,A6,FRECHET:7,TOPREAL9:13;
hence thesis by A7;
end;
then f1 is open by Th6;
hence thesis by A1,Th1;
end;
begin :: Continuous functions
theorem
for f being Function of T,TopSpaceMetr(M) holds f is continuous iff
for p being Point of T, q being Point of M, r being positive (real number)
st q = f.p
ex W being open Subset of T st p in W & f.:W c= Ball(q,r)
proof
let f be Function of T,TopSpaceMetr(M);
thus f is continuous implies
for p being Point of T, q being Point of M, r being positive (real number)
st q = f.p
ex W being open Subset of T st p in W & f.:W c= Ball(q,r)
proof
assume
A1: f is continuous;
let p be Point of T;
let q be Point of M;
let r be positive (real number);
assume
A2: f.p = q;
reconsider V = Ball(q,r) as Subset of TopSpaceMetr(M);
A3: q in Ball(q,r) by GOBOARD6:4;
V is open by TOPMETR:21;
then ex W being Subset of T st p in W & W is open & f.:W c= V
by A1,A2,A3,JGRAPH_2:20;
hence thesis;
end;
assume
A4: for p being Point of T, q being Point of M,
r being positive (real number) st q = f.p
ex W being open Subset of T st p in W & f.:W c= Ball(q,r);
for p being Point of T, V being Subset of TopSpaceMetr(M)
st f.p in V & V is open holds
ex W being Subset of T st p in W & W is open & f.:W c= V
proof
let p be Point of T, V be Subset of TopSpaceMetr(M) such that
A5: f.p in V;
reconsider u = f.p as Point of M;
assume V is open;
then Int V = V by TOPS_1:55;
then consider e being real number such that
A6: e > 0 & Ball(u,e) c= V by A5,GOBOARD6:7;
ex W being open Subset of T st p in W & f.:W c= Ball(u,e) by A4,A6;
hence thesis by A6,XBOOLE_1:1;
end;
hence thesis by JGRAPH_2:20;
end;
theorem
for f being Function of TopSpaceMetr(M),T holds f is continuous iff
for p being Point of M, V being open Subset of T st f.p in V
ex s being positive (real number) st f.:Ball(p,s) c= V
proof
let f be Function of TopSpaceMetr(M),T;
hereby
assume
A1: f is continuous;
let p be Point of M;
let V be open Subset of T;
assume f.p in V;
then consider W being Subset of TopSpaceMetr(M) such that
A2: p in W and
A3: W is open and
A4: f.:W c= V by A1,JGRAPH_2:20;
Int W = W by A3,TOPS_1:55;
then consider s being real number such that
A5: s > 0 and
A6: Ball(p,s) c= W by A2,GOBOARD6:7;
reconsider s as positive (real number) by A5;
take s;
f.:Ball(p,s) c= f.:W by A6,RELAT_1:156;
hence f.:Ball(p,s) c= V by A4,XBOOLE_1:1;
end;
assume
A7: for p being Point of M, V being open Subset of T st f.p in V
ex s being positive (real number) st f.:Ball(p,s) c= V;
for p being Point of TopSpaceMetr(M), V being Subset of T
st f.p in V & V is open holds
ex W being Subset of TopSpaceMetr(M) st p in W & W is open & f.:W c= V
proof
let p be Point of TopSpaceMetr(M), V be Subset of T such that
A8: f.p in V and
A9: V is open;
reconsider u = p as Point of M;
consider s being positive (real number) such that
A10: f.:Ball(u,s) c= V by A7,A8,A9;
reconsider W = Ball(u,s) as Subset of TopSpaceMetr(M);
take W;
thus p in W by GOBOARD6:4;
thus W is open by TOPMETR:21;
thus thesis by A10;
end;
hence thesis by JGRAPH_2:20;
end;
theorem Th17:
for f being Function of TopSpaceMetr(M1),TopSpaceMetr(M2) holds
f is continuous iff
for p being Point of M1, q being Point of M2, r being positive (real number)
st q = f.p
ex s being positive (real number) st f.:Ball(p,s) c= Ball(q,r)
proof
let f be Function of TopSpaceMetr(M1),TopSpaceMetr(M2);
hereby
assume
A1: f is continuous;
let p be Point of M1;
let q be Point of M2;
let r be positive (real number);
assume that
A2: q = f.p;
r in REAL by XREAL_0:def 1;
then consider s being Element of REAL such that
A3: s > 0 and
A4: for w being Element of M1, w1 being Element of M2 st
w1 = f.w & dist(p,w) < s holds dist(q,w1) < r by A1,A2,UNIFORM1:5;
reconsider s as positive (real number) by A3;
take s;
thus f.:Ball(p,s) c= Ball(q,r)
proof
let y be set;
assume y in f.:Ball(p,s);
then consider x being Point of TopSpaceMetr(M1) such that
A5: x in Ball(p,s) and
A6: f.x = y by FUNCT_2:116;
reconsider x1 = x as Point of M1;
reconsider y1 = y as Point of M2 by A6;
dist(p,x1) < s by A5,METRIC_1:12;
then dist(q,y1) < r by A6,A4;
hence y in Ball(q,r) by METRIC_1:12;
end;
end;
assume
A7: for p being Point of M1, q being Point of M2,
r being positive (real number) st q = f.p
ex s being positive (real number) st f.:Ball(p,s) c= Ball(q,r);
for r being real number, u being Element of M1, u1 being Element of M2
st r > 0 & u1 = f.u
ex s being real number st s > 0 & for w being Element of M1,
w1 being Element of M2 st w1 = f.w & dist(u,w)~~ 0 & u1 = f.u;
then consider s being positive (real number) such that
A8: f.:Ball(u,s) c= Ball(u1,r) by A7;
take s;
thus s > 0;
let w be Element of M1, w1 be Element of M2 such that
A9: w1 = f.w;
assume dist(u,w) < s;
then w in Ball(u,s) by METRIC_1:12;
then f.w in f.:Ball(u,s) by FUNCT_2:43;
hence thesis by A8,A9,METRIC_1:12;
end;
hence thesis by UNIFORM1:4;
end;
theorem
for f being Function of T,TOP-REAL m holds f is continuous iff
for p being Point of T, r being positive (real number)
ex W being open Subset of T st p in W & f.:W c= Ball(f.p,r)
proof
let f be Function of T,TOP-REAL m;
A1: m in NAT by ORDINAL1:def 13;
thus f is continuous implies
for p being Point of T, r being positive (real number)
ex W being open Subset of T st p in W & f.:W c= Ball(f.p,r)
proof
assume
A2: f is continuous;
let p be Point of T;
let r be positive (real number);
f.p in Ball(f.p,r) by A1,TOPGEN_5:17;
then ex W being Subset of T st p in W & W is open & f.:W c= Ball(f.p,r)
by A2,JGRAPH_2:20;
hence thesis;
end;
assume
A3: for p being Point of T, r being positive (real number)
ex W being open Subset of T st p in W & f.:W c= Ball(f.p,r);
for p being Point of T, V being Subset of TOP-REAL m
st f.p in V & V is open holds
ex W being Subset of T st p in W & W is open & f.:W c= V
proof
let p be Point of T, V be Subset of TOP-REAL m such that
A4: f.p in V;
reconsider u = f.p as Point of Euclid m by EUCLID:71;
assume V is open;
then Int V = V by TOPS_1:55;
then consider e being real number such that
A5: e > 0 and
A6: Ball(u,e) c= V by A1,A4,GOBOARD6:8;
A7: Ball(u,e) = Ball(f.p,e) by A1,TOPREAL9:13;
ex W being open Subset of T st p in W & f.:W c= Ball(f.p,e) by A3,A5;
hence thesis by A6,XBOOLE_1:1,A7;
end;
hence thesis by JGRAPH_2:20;
end;
theorem
for f being Function of TOP-REAL m,T holds f is continuous iff
for p being Point of TOP-REAL m, V being open Subset of T st f.p in V
ex s being positive (real number) st f.:Ball(p,s) c= V
proof
let f be Function of TOP-REAL m,T;
A1: m in NAT by ORDINAL1:def 13;
hereby
assume
A2: f is continuous;
let p be Point of TOP-REAL m;
let V be open Subset of T;
assume f.p in V;
then consider W being Subset of TOP-REAL m such that
A3: p in W and
A4: W is open and
A5: f.:W c= V by A2,JGRAPH_2:20;
reconsider u = p as Point of Euclid m by EUCLID:71;
Int W = W by A4,TOPS_1:55;
then consider s being real number such that
A6: s > 0 and
A7: Ball(u,s) c= W by A1,A3,GOBOARD6:8;
reconsider s as positive (real number) by A6;
take s;
Ball(u,s) = Ball(p,s) by A1,TOPREAL9:13;
then f.:Ball(p,s) c= f.:W by A7,RELAT_1:156;
hence f.:Ball(p,s) c= V by A5,XBOOLE_1:1;
end;
assume
A8: for p being Point of TOP-REAL m, V being open Subset of T st f.p in V
ex s being positive (real number) st f.:Ball(p,s) c= V;
for p being Point of TOP-REAL m, V being Subset of T
st f.p in V & V is open holds
ex W being Subset of TOP-REAL m st p in W & W is open & f.:W c= V
proof
let p be Point of TOP-REAL m, V be Subset of T such that
A9: f.p in V and
A10: V is open;
consider s being positive (real number) such that
A11: f.:Ball(p,s) c= V by A8,A9,A10;
take W = Ball(p,s);
thus p in W by A1,TOPGEN_5:17;
thus W is open;
thus thesis by A11;
end;
hence thesis by JGRAPH_2:20;
end;
theorem
for f being Function of TOP-REAL m, TOP-REAL n holds f is continuous iff
for p being Point of TOP-REAL m, r being positive (real number)
ex s being positive (real number) st f.:Ball(p,s) c= Ball(f.p,r)
proof
let f be Function of TOP-REAL m, TOP-REAL n;
A1: the TopStruct of TOP-REAL m = TopSpaceMetr Euclid m &
the TopStruct of TOP-REAL n = TopSpaceMetr Euclid n by EUCLID:def 8;
then reconsider f1 = f as Function of TopSpaceMetr Euclid m,
TopSpaceMetr Euclid n;
A2: m in NAT & n in NAT by ORDINAL1:def 13;
hereby
assume
A3: f is continuous;
let p be Point of TOP-REAL m;
let r be positive (real number);
reconsider p1 = p as Point of Euclid m by EUCLID:71;
reconsider q1 = f.p as Point of Euclid n by EUCLID:71;
f1 is continuous by A1,A3,YELLOW12:36;
then consider s being positive (real number) such that
A4: f1.:Ball(p1,s) c= Ball(q1,r) by Th17;
take s;
Ball(p1,s) = Ball(p,s) & Ball(q1,r) = Ball(f.p,r) by A2,TOPREAL9:13;
hence f.:Ball(p,s) c= Ball(f.p,r) by A4;
end;
assume
A5: for p being Point of TOP-REAL m, r being positive (real number)
ex s being positive (real number) st f.:Ball(p,s) c= Ball(f.p,r);
for p being Point of Euclid m, q being Point of Euclid n,
r being positive (real number) st q = f1.p
ex s being positive (real number) st f1.:Ball(p,s) c= Ball(q,r)
proof
let p be Point of Euclid m, q be Point of Euclid n,
r be positive (real number) such that
A6: q = f1.p;
reconsider p1 = p as Point of TOP-REAL m by EUCLID:71;
consider s being positive (real number) such that
A7: f.:Ball(p1,s) c= Ball(f.p1,r) by A5;
take s;
Ball(p1,s) = Ball(p,s) & Ball(f.p1,r) = Ball(q,r) by A2,A6,TOPREAL9:13;
hence thesis by A7;
end;
then f1 is continuous by Th17;
hence thesis by A1,YELLOW12:36;
end;
theorem
for f being Function of T,R^1 holds f is continuous iff
for p being Point of T, r being positive (real number)
ex W being open Subset of T st p in W & f.:W c= ].f.p-r,f.p+r.[
proof
let f be Function of T,R^1;
thus f is continuous implies
for p being Point of T, r being positive (real number)
ex W being open Subset of T st p in W & f.:W c= ].f.p-r,f.p+r.[
proof
assume
A1: f is continuous;
let p be Point of T;
let r be positive (real number);
A2: f.p in ].f.p-r,f.p+r.[ by TOPREAL6:20;
R^1(].f.p-r,f.p+r.[) is open;
then ex W being Subset of T st p in W & W is open &
f.:W c= ].f.p-r,f.p+r.[ by A1,A2,JGRAPH_2:20;
hence thesis;
end;
assume
A3: for p being Point of T, r being positive (real number)
ex W being open Subset of T st p in W & f.:W c= ].f.p-r,f.p+r.[;
for p being Point of T, V being Subset of R^1 st f.p in V & V is open holds
ex W being Subset of T st p in W & W is open & f.:W c= V
proof
let p be Point of T, V be Subset of R^1 such that
A4: f.p in V;
assume V is open;
then consider r being Element of REAL such that
A5: r > 0 and
A6: ].f.p-r,f.p+r.[ c= V by A4,FRECHET:8;
ex W being open Subset of T st p in W & f.:W c= ].f.p-r,f.p+r.[ by A3,A5;
hence thesis by A6,XBOOLE_1:1;
end;
hence thesis by JGRAPH_2:20;
end;
theorem
for f being Function of R^1,T holds f is continuous iff
for p being Point of R^1, V being open Subset of T st f.p in V
ex s being positive (real number) st f.:].p-s,p+s.[ c= V
proof
let f be Function of R^1,T;
hereby
assume
A1: f is continuous;
let p be Point of R^1;
let V be open Subset of T;
assume f.p in V;
then consider W being Subset of R^1 such that
A2: p in W and
A3: W is open and
A4: f.:W c= V by A1,JGRAPH_2:20;
consider s being Element of REAL such that
A5: s > 0 and
A6: ].p-s,p+s.[ c= W by A2,A3,FRECHET:8;
reconsider s as positive (real number) by A5;
take s;
f.:].p-s,p+s.[ c= f.:W by A6,RELAT_1:156;
hence f.:].p-s,p+s.[ c= V by A4,XBOOLE_1:1;
end;
assume
A7: for p being Point of R^1, V being open Subset of T st f.p in V
ex s being positive (real number) st f.:].p-s,p+s.[ c= V;
for p being Point of R^1, V being Subset of T st f.p in V & V is open holds
ex W being Subset of R^1 st p in W & W is open & f.:W c= V
proof
let p be Point of R^1, V be Subset of T such that
A8: f.p in V and
A9: V is open;
consider s being positive (real number) such that
A10: f.:].p-s,p+s.[ c= V by A7,A8,A9;
take W = R^1(].p-s,p+s.[);
thus p in W by TOPREAL6:20;
thus W is open;
thus thesis by A10;
end;
hence thesis by JGRAPH_2:20;
end;
theorem
for f being Function of R^1,R^1 holds f is continuous iff
for p being Point of R^1, r being positive (real number)
ex s being positive (real number) st f.:].p-s,p+s.[ c= ].f.p-r,f.p+r.[
proof
let f be Function of R^1,R^1;
hereby
assume
A1: f is continuous;
let p be Point of R^1;
let r be positive (real number);
reconsider p1 = p, q1 = f.p as Point of RealSpace;
consider s being positive (real number) such that
A2: f.:Ball(p1,s) c= Ball(q1,r) by A1,Th17;
take s;
p1 in REAL & q1 in REAL & r in REAL & s in REAL by XREAL_0:def 1;
then Ball(p1,s) = ].p1-s,p1+s.[ & Ball(q1,r) = ].q1-r,q1+r.[
by FRECHET:7;
hence f.:].p-s,p+s.[ c= ].f.p-r,f.p+r.[ by A2;
end;
assume
A3: for p being Point of R^1, r being positive (real number)
ex s being positive (real number) st f.:].p-s,p+s.[ c= ].f.p-r,f.p+r.[;
for p, q being Point of RealSpace, r being positive (real number)
st q = f.p
ex s being positive (real number) st f.:Ball(p,s) c= Ball(q,r)
proof
let p, q be Point of RealSpace,
r be positive (real number) such that
A4: q = f.p;
consider s being positive (real number) such that
A5: f.:].p-s,p+s.[ c= ].f.p-r,f.p+r.[ by A3;
take s;
p in REAL & q in REAL & r in REAL & s in REAL by XREAL_0:def 1;
then Ball(p,s) = ].p-s,p+s.[ & Ball(q,r) = ].q-r,q+r.[ by FRECHET:7;
hence thesis by A5,A4;
end;
hence f is continuous by Th17;
end;
theorem
for f being Function of TOP-REAL m,R^1 holds f is continuous iff
for p being Point of TOP-REAL m, r being positive (real number)
ex s being positive (real number) st f.:Ball(p,s) c= ].f.p-r,f.p+r.[
proof
let f be Function of TOP-REAL m, R^1;
A1: the TopStruct of TOP-REAL m = TopSpaceMetr Euclid m by EUCLID:def 8;
then reconsider f1 = f as Function of TopSpaceMetr Euclid m,R^1;
A2: m in NAT by ORDINAL1:def 13;
hereby
assume
A3: f is continuous;
let p be Point of TOP-REAL m;
let r be positive (real number);
reconsider p1 = p as Point of Euclid m by EUCLID:71;
reconsider q1 = f.p as Point of RealSpace;
f1 is continuous by A1,A3,YELLOW12:36;
then consider s being positive (real number) such that
A4: f.:Ball(p1,s) c= Ball(q1,r) by A1,Th17;
take s;
q1 in REAL & r in REAL by XREAL_0:def 1;
then Ball(p1,s) = Ball(p,s) & Ball(q1,r) = ].f.p-r,f.p+r.[
by A2,TOPREAL9:13,FRECHET:7;
hence f.:Ball(p,s) c= ].f.p-r,f.p+r.[ by A4;
end;
assume
A5: for p being Point of TOP-REAL m, r being positive (real number)
ex s being positive (real number) st f.:Ball(p,s) c= ].f.p-r,f.p+r.[;
for p being Point of Euclid m, q being Point of RealSpace,
r being positive (real number) st q = f1.p
ex s being positive (real number) st f1.:Ball(p,s) c= Ball(q,r)
proof
let p be Point of Euclid m, q be Point of RealSpace,
r be positive (real number) such that
A6: q = f1.p;
reconsider p1 = p as Point of TOP-REAL m by EUCLID:71;
consider s being positive (real number) such that
A7: f.:Ball(p1,s) c= ].f.p-r,f.p+r.[ by A5;
take s;
q in REAL & r in REAL by XREAL_0:def 1;
then Ball(p1,s) = Ball(p,s) & ].f.p-r,f.p+r.[ = Ball(q,r)
by A2,A6,TOPREAL9:13,FRECHET:7;
hence thesis by A7;
end;
then f1 is continuous by Th17;
hence thesis by A1,YELLOW12:36;
end;
theorem
for f being Function of R^1,TOP-REAL m holds f is continuous iff
for p being Point of R^1, r being positive (real number)
ex s being positive (real number) st f.:].p-s,p+s.[ c= Ball(f.p,r)
proof
let f be Function of R^1, TOP-REAL m;
A1: the TopStruct of TOP-REAL m = TopSpaceMetr Euclid m by EUCLID:def 8;
then reconsider f1 = f as Function of R^1,TopSpaceMetr Euclid m;
A2: m in NAT by ORDINAL1:def 13;
hereby
assume
A3: f is continuous;
let p be Point of R^1;
let r be positive (real number);
reconsider p1 = p as Point of RealSpace;
reconsider q1 = f.p as Point of Euclid m by EUCLID:71;
f1 is continuous by A1,A3,YELLOW12:36;
then consider s being positive (real number) such that
A4: f1.:Ball(p1,s) c= Ball(q1,r) by Th17;
take s;
p in REAL & s in REAL by XREAL_0:def 1;
then Ball(p1,s) = ].p-s,p+s.[ & Ball(q1,r) = Ball(f.p,r)
by A2,TOPREAL9:13,FRECHET:7;
hence f.:].p-s,p+s.[ c= Ball(f.p,r) by A4;
end;
assume
A5: for p being Point of R^1, r being positive (real number)
ex s being positive (real number) st f.:].p-s,p+s.[ c= Ball(f.p,r);
for p being Point of RealSpace, q being Point of Euclid m,
r being positive (real number) st q = f.p
ex s being positive (real number) st f.:Ball(p,s) c= Ball(q,r)
proof
let p be Point of RealSpace, q be Point of Euclid m,
r be positive (real number) such that
A6: q = f.p;
reconsider p1 = p as Point of R^1;
consider s being positive (real number) such that
A7: f.:].p-s,p+s.[ c= Ball(f.p1,r) by A5;
take s;
p in REAL & s in REAL by XREAL_0:def 1;
then ].p-s,p+s.[ = Ball(p,s) & Ball(f.p1,r) = Ball(q,r)
by A2,A6,TOPREAL9:13,FRECHET:7;
hence thesis by A7;
end;
then f1 is continuous by A1,Th17;
hence thesis by A1,YELLOW12:36;
end;
~~